Marcus-Lushnikov processes, Smoluchowski's and Flory's models

TL;DR

This paper investigates the Marcus-Lushnikov process with a strongly gelling kernel, analyzing its hydrodynamic limits and the emergence of giant particles, and compares the Smoluchowski and Flory models in this context.

Contribution

It demonstrates that limits of the process with a cut-off kernel solve either the Smoluchowski or Flory equations, and characterizes the asymptotic behavior of the largest particle without cut-off.

Findings

Limits solve Smoluchowski or Flory equations with cut-off.

Single giant particle emerges without cut-off, representing mass loss.

Asymptotic behavior of the largest particle analyzed.

Abstract

The Marcus-Lushnikov process is a finite stochastic particle system in which each particle is entirely characterized by its mass. Each pair of particles with masses $x$ and $y$ merges into a single particle at a given rate $K(x,y)$. We consider a {\it strongly gelling} kernel behaving as $K(x,y)=x^\alpha y + x y^\alpha$ for some $\alpha\in (0,1]$. In such a case, it is well-known that {\it gelation} occurs, that is, giant particles emerge. Then two possible models for hydrodynamic limits of the Marcus-Lushnikov process arise: the Smoluchowski equation, in which the giant particles are inert, and the Flory equation, in which the giant particles interact with finite ones. We show that, when using a suitable cut-off coagulation kernel in the Marcus-Lushnikov process and letting the number of particles increase to infinity, the possible limits solve either the Smoluchowski equation or the Flory equation. We also study the asymptotic behaviour of the largest particle in the Marcus-Lushnikov process without cut-off and show that there is only one giant particle. This single giant particle represents, asymptotically, the lost mass of the solution to the Flory equation.